Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, p\neq 0$. $\dfrac{{(z^{3}p^{3})^{-3}}}{{(z^{5}p^{-3})^{-1}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{3}p^{3})^{-3} = (z^{3})^{-3}(p^{3})^{-3}}$ On the left, we have ${z^{3}}$ to the exponent ${-3}$ . Now ${3 \times -3 = -9}$ , so ${(z^{3})^{-3} = z^{-9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{3}p^{3})^{-3}}}{{(z^{5}p^{-3})^{-1}}} = \dfrac{{z^{-9}p^{-9}}}{{z^{-5}p^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-9}p^{-9}}}{{z^{-5}p^{3}}} = \dfrac{{z^{-9}}}{{z^{-5}}} \cdot \dfrac{{p^{-9}}}{{p^{3}}} = z^{{-9} - {(-5)}} \cdot p^{{-9} - {3}} = z^{-4}p^{-12}$